Gravitational Waves Gravitons Vishal Kasliwal Classical Electromagnetism Vacuum

Gravitational Waves (& Gravitons ? ) -Vishal Kasliwal

Classical Electromagnetism Vacuum Maxwell Field Equations ! t! h g i L Electromagnetic waves

Quantum Electromagnetism Hamiltonian of Quantized Field Particles ! s! n o t Pho where (Harmonic oscillators !!) Eigenstates and Energy is the occupation number of the mode a ie. we have n photons with momentum k. Photons are spin one Bosons because they arise from the quantization of a vector field.

General Relativity Lorentz Transforms Lorentz matrices These replace Galilean Transforms Geodesics Einstein Field Equation When no ‘forces’ are acting on a body, it travels along a geodesic! Fancy Newton’s 1 st Law ‘Force’ = Geometry ‘Or how we create gravity!’ – Replaces Newton’s Law of Universal Gravitation • Set of 10 coupled, non-linear, hyperbolic-elliptical PDEs • Linearize • Weak field, low velocity limit curved part flat part small perturbation function of small perturbation

Classical Gravity Vacuum Einstein Field Equations t? igh L G Weak field – low velocity Linearized Gravitational waves Do They Exist? • 1993 Noble Prize – Russell Hulse & John Taylor • PSR B 1913+16: Binary star with pulsar as one component. • Strong emitter of gravitational radiation. • Orbit decays as predicted by loss of energy due to G-waves in accordance with GR

Gravitational Waves + polarization × polarization • Amplitude (h): Fraction of stretching or squeezing. • Frequency (n): 1/Time between two max squeezes. • Wavelength (l): Distance along wave between two max squeezes. • Speed (c): Speed of light. Produced by • Binary systems with neutron stars or black holes • Supernova Detected by • Weber Bar type detectors: 60 s, 70 s but still investigated – Mini. Grail (The Netherlands) • Interferometric type: LIGO (US), CLIO (Japan), GEO 600 (Germany), VIRGO (Europe – Italy), TAMA 300 (Japan)

Gravitons • Quantization of gravitational field (hasn’t been done yet) produces gravitons. • Massless tensor-gauge bosons. • Carry very little energy – might be impossible to detect. • G-wave is wave packet of gravitons just as EM-wave is wave packet of photons. • Polarization of gravitational waves invariant under rotations of 180 degrees. • Spin = 360/(Rotation angle under which polarization modes are invariant) = 360/180 = 2 • Applies to EM waves (S = 360/360 = 1) • Not much known – more research required!

Fin

Planck Units • Length & Time • Energy & Mass • Simplifies forms of equations.

Quantized Hamiltonian of EM Field Oscillator Hamiltonian Raising Operator

Metric

Curvature I Christoffel Symbol Covariant Derivative Reimann Curvature Tensor Ricci Tensor Parallel Transport ! Scalar Curvature Einstein Tensor

Curvature II Tensor Symbol Indep. Comps Reimann 20 Ricci 10 Scalar Curvature 1 Einstein 10 Geodesics Cosmological Constant Vacuum energy of free space. Einstein Tensor Measures volume distortions Energy-Momentum Tensor Typically derived from action principle or equation of motion

Weber Detector Mini. Grail

Interferometric Detector